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SYLLABUS 


BOURSE  IN  PLANE  ANALYTIC  GEOMETRY. 


BOSTON: 
PUBLISHED  MY  GINN,  HEATH,  &  CO. 

1883. 


COPVRIOIITEO  BY  OiNN,  Ueath,  &  Co.,  1883. 


Mathematical  Books. 


Intro. 
Price. 

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HALSTED:  Mensuration 80 

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PEIEOE :       Elements  of  Logarithms 40 

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Elements  of  Algebra 90 

Complete  Algebra 1.12 

Plane  Geometry .60 

Plane  and  Solid  Geometry 1.00 

Plane  Trigonometry.     Paper     ......      .30 

Plane  and  Spherical  Trigonometry 48 

Plane  Trigonometry  and  Tables.  Paper ...  .48 
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Practical  Arithmetic 75 

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GINN,  HEATH  &  CO.,  Publishers. 

BOSTON.       NEW  YORK.       CHICAGO. 


3V  A 


SYLLABUS 


COUESE  IN  PLANE  ANALYTIC  GEOMETRY. 

V  1.   Explain  how,  in  appWing  Algebra  to  Geometry,  difference 

I         of  sign  may  be  interpreted  geometrically.     To  illustrate  the 

*^  gain  in  generality  arising  from  the  use  of  negative  as  well  as 
positive  quantities,  prove  that,  with  the  convention  that  opposite 
signs  indicate  opposite  directions,  the  distance  of  two  points 
apart  is  always  equal  to  the  difference  between  their  distances 

^"t       from  any  third  point  on  the  line  through  them. 


^ 
^ 


2.  Explain  the  method  of  representing  the  position  of  a 
p>oint  in  a  plane  by  Cartesian  coordinates.  Define  axes, 
origin,  abscissa,  ordinate,  coordinates,  axis  of  abscissas,  axis  of 
ordinates.  Distinguish  between  oblique  and  rectangular  coordi- 
nates.    Give  the  convention  concerning  the  signs  of  coordinates. 

3.  Problem.  To  find  the  distance  between  two  points  in 
terms  of  then:  coordinates. 

[1]     D  =  V(aij  —  a;i)2+  (^2—  ^i)*  +  ^  {^2  —  a-':)  (y.  —  y^)  cos  w. 
[2]  \:D  =  ^{x,-x,y+{y2-y,y].* 

^^         4.   Problem.     To  divide  a  line  in  any  given  ratio  mi :  m.2. 

rQT     ^     m^Xi  +  m^x^  Wj.Vi  +  Wiya 

-"  wio-f  mi  m^-^-mi 

If  the  line  is  bisected, 

Xi  +  x^  y,  +  ?/2 

^  [4]     ^=-^-'  y  =  ^— 

•  Brackets  indicate  rectangular  coordinates. 


V 


4G2G18 


5.  Explain  how  an  equation  containing  two  variables  x  and  y 
can  represent  a  curve. 

Define  the  locus  of  an  equation  ;  the  equation  of  a  curve.  If 
a  curve  is  given  by  its  equation  and  a  point  by  its  coordinates, 
show  how  it  can  be  ascertained  whether  the  point  does  or  does 
not  lie  on  the  curve. 

Explain  the  method  of  constructing  the  curve  represented  b}'^ 
a  given  equation,  b}'  finding  and  plotting  points  whose  coordi- 
nates satisf}^  the  equation. 

Define  the  intercepts  of  a  curv^e  on  the  axes,  and  show  how 
they  may  be  found  from  the  equation  of  the  curve. 

If  the  equation  contains  no  constant  term,  the  curve  passes 
through  the  origin. 

6.  If  two  curves  are  given  by  their  equations,  show  how  to 
find  the  coordinates  of  their  points  of  intersection. 

If  w  =  0  and  V  =  0  are  the  equations  of  two  curves,  prove  that 
u-\-kv  =  0  represents  a  curve  passing  through  all  their  points 
of  intersection  and  having  no  other  point  in  common  with  either 
of  them. 

Show  that  uv  =  0  represents  both  of  the  curves  w  =  0  and 
v  =  0. 

The  Straight  Line. 

7.  Find  the  equation  of  a  line  when  its  intercepts  a  and  6 
axe  given ; 

[5]   ^+1=1; 

a     0 
when  the  intercept  h  on  the  axis  of  ordinates  and  the  angle  y 
made  with  the  axis  of  abscissas  are  given  ; 

[61     y=lx-\-h       where      Z  =  ^— — "^ — ^^     [=tany]; 
when  a  point  (a'l,  y^)  of  the  line  and  y  are  given  ; 

[7]    y-yi  =  i{x-x{)', 
when  two  points  of  the  line  are  given ; 


[8]      i^IllL  =  f^Il^; 

Vi  —  Vx         ^2         •''1 

when  the  length  of  the  perpendicular  p  from  the  origin  upon  the 

line  and  the  angle  a  which  p  makes  with  the  axis  of  abscissas 

are  given ; 

[9]     xcosa  +  y  cos(w  —  a)  =  p  ; 

[10]   [a;cosa  +  ?/sina=p]. 

8.  Prove  that  every  equation  of  the  first  degree  Ax  +  By  ■\- 
C=0  represents  a  straight  line,  and  find  values  for  a,  6,  and  /, 
in  terms  of  the  coefficients  of  the  equation. 

[11]     a  =  -f,       6  =  -|       l^-± 

9.  Show  how  to  reduce  Aic  -\-By  +  C=0  to  the  form  x  cos  a 
+  y  8ina=2),  the  coordinates  being  rectangular. 

[12]      -7=^^  +  -tJ=2/  =  —     ^ 


V^T^        V^'  +  Jy  -VA'  +  B' 

10.  Find  the  angle  between  two  straight  lines  whose  equa- 
tions are  given  in  rectangular  coordinates. 

•-     -^     \_  l  +  tl,     AA,  +  BB, 

If  the  lines  are  parallel, 

[14]     l,  =  l         or    ^^^. 
'-     -■      '  Bi      B 

If  the  lines  are  perpendicular, 

[15]     ;.  =  --     or    ^  =  --. 

11 .  Problem.  To  find  the  equation  of  a  line  passing  through 
a  given  point  and  parallel  to  a  given  line  ;  perpendicular  to  a 
given  line.     Use  the  method  of  undetermined  coefficients. 

[16]     \_Ax  +  By=Ax^  +  By,-\. 
[17]     iBx-Ay  =  Bx^-Ay{\, 


12.   Problem.     To  find  the  distance  from  a  given  point  to  a 
given  line. 


[18] 


D  = 


^A'+B'    J 


13.  Problem.     To  find  the  area  of  a  triangle  whose  vertices 
are  given. 

[19]     [M=i\yi(x2—X;i)+y2(xs-Xi)+ys{xi-X2)l^. 

14.  Describe  the  general  method  of  solving  problems  in  loci. 


Transformation  op  Coordinates. 


15.  De&ne  orthogonal  projection,  parallel  projection.  Obtain 
the  projection  of  a  line  a  upon  each  axis,  the  projection  being 
made  by  the  aid  of  lines  parallel  to  the  other  axis. 


[20]     a^=a- 


sm  1  ^  — 


a,.  =  a- 


sin 


16.   Obtain  formulas  for  transforming  from  a  given  set  of 
axes  to  any  second  set. 


fX=  Xo-{-  x' 


sin   -^  — 
\x      X 


'y     V 
sin  [  ^  — ^ 

,X      X 


[21] 


sin' 


sin 


'<y  =  yo+x' 


sin 


X 

x'  .   y' 

sm^ 

X  ,         X 


sin 


Find  the  reduced  forms  of   [21]  when  the  new  axes  are 
parallel  to  the  old. 


[22]     x  =  Xo  +  x',        2/ =2/0+2/'. 


5 

17.  Show  that  the  degree  of  an  equation  cannot  be  altered 
b}'  any  transformation  of  coordinates.  The  degree  of  an  equa- 
tion shows  the  number  of  points  in  which  the  curve  can  be  cut 
by  a  straight  line. 

18.  Explain  jwlar  coordinates.  Define  the  initial  line.  Find 
formulas  for  transforming  from  Cartesian  to  polar  coordinates. 

[23]     ^^,^i»(— <^),    y^r^^. 
smo)  suiw 

[24]   [a;  =  rcos^,  ?/  =  rsiu(^]. 


The  Circle. 

19.  Find  the  equation  of  a  circle  in  terms  of  the  coordinates 
of  the  centre  and  the  length  of  the  radius. 

[25]      {x-ay-+{y-hy  =  i^. 

When  the  centre  is  at  the  origin,  the  equation  reduces  to 

[26]     x2  +  2/-  =  r2. 

20.  Show  that  the  most  general  expanded  form  of  the  equa- 
tion of  a  circle  is 

[27]     IT  4- y-  +'Px  +  Ey  +  F=(i', 

and  by  comparing  this  with  the  general  equation  of  the  second 
degree,  Ax^ -^-Bxy  ^Cy- +  Dx-{- Ey -\- F=Q^  obtain  the  con- 
ditions that  an  equation  of  the  second  degree  shall  represent 
a  circle. 

[28]     A=C,        B  =  0. 

Show  how  to  reduce  an  equation  in  the  form  [27]  to  the 
form  [25],  and  thus  to  obtain  the  centre  and  radius. 

Prove  that  every  e(iuation  of  the  form  [27]  represents  a 
circle,  real,  null,  or  imaginary. 


6 

21.  Problem.  To  find  the  equation  of  a  circle  passing 
through  three  given  points. 

22.  Problem.  To  find  the  equation  of  the  common  chord  of 
two  circles  whose  equations  are  given. 

23.  Find  the  equation  of  a  tangent  at  a  given  point  on  the 
circumference  of  a  circle. 

[29]     Xj^x+yiy  =  r^. 

[30]     (x,  -a)(x-  a)  +  (y,  -b)(y-b)  =  /-'. 

24.  Problem.  To  find  the  equation  of  tangents  from  a 
given  point  to  a  circle. 

25.  Find  tlie  locus  of  points  dividing  harmonically  secants 
drawn  through  (x^,  yi). 

[31]     XiX-^yiy  =  t^. 

This  is  called  the  polar  of  (xi,  y^) . 

If  (a^,  2/i)  is  on  the  circle,  its  polar  is  the  tangent  at  (a^i,  yi). 

Find  the  coordinates  of  the  pole  of  Ax  +  By  +  C=0. 

Show  that  the  polar  of  a  point  is  perpendicular  to  the  line 
joining  the  point  with  the  centre  of  the  circle ;  and  that  the 
product  of  the  distance  of  the  point  from  the  centre  by  the 
distance  of  the  polar  from  the  centi'e  is  equal  to  the  square  of 
the  radius. 

26.  Show  that  if  {xi,  y^  is  without  the  circle,  its  polar 
passes  through  the  points  of  contact  of  tangents  from  (ccj,  y^) 
to  the  circle. 

27.  Prove  that  if  a  set  of  points  lie  on  a  line,  their  polars  all 
pass  through  the  pole  of  that  line ;  and,  conversely,  that  if  a 
set  of  lines  pass  through  a  point,  theii'  poles  lie  on  the  polar  of 
that  point. 


28.  Find  the  locus  of  the  middle  points  of  a  set  of  parallel 
chords. 

[32]     x  +  ytiin6  =  0. 

Such  a  line  is  called  a  diameter. 

Prove  that  every  chord  through  the  centre  is  a  diameter. 

TuE  Conic  Sections. 

29.  Define  a  Conic  Section,  and  distinguish  between  the 
Ellijise,  the  Hyperbola,  and  the  Parabola.  Show  how  to  con- 
struct these  curves  from  their  definitions.  Define  the  focus; 
the  directrix;  the  transverse  axis;  the  centre.  Show  that  the 
Parabola  has  no  centre. 

30.  Obtain  the  equation  of  a  conic,  taking  the  directrix  and 
the  transverse  axis  as  axes  of  coordinates. 

[33]     {l-e-)x^-{-y-—2mx  +  m-  =  0. 

Find  the  intercepts  on  the  axis  of  abscissas,  and  obtain  from 
them  the  coordinates  of  the  centre.  Transform  to  the  centre  as 
origin ;  find  the  intercepts  on  the  new  axes,  and,  representing 
them  by  a  and  &,  express  the  equation  of  a  central  conic  in 
terms  of  them. 

a-      b- 
The  equation  of  the  Hyperbola  is  usually  written 

[35]     ^-t=i, 
•-     -■     a;'     b- 

Define  conjugate  hyperbolas;  the  equilateral  hyperbola. 

Prove  that  each  axis  of  a  central  conic  bisects  all  the  chords 
parallel  to  the  other. 

Obtain  the  equation  of  the  Parabola  referred  to  its  transverse 
axis  and  the  tangent  at  the  vertex  as  axes. 

[3G]     if  =  2mx. 


8 

31.  Transform  the  equations  of  the  Ellipse  and  the  Hyper- 
bola to  polar  coordinates,  and  then  investigate  the  forms  of  the 
curves.  Prove  that  every  chord  through  the  centre  is  bisected 
by  the  centre. 

Define  the  asymptotes  of  the  H3'perbola.  Show  how  to 
construct  them  when  the  semi-axes  are  known.  Find  their 
equations. 

[37]     -  +  ^  =  0,  --^  =  0. 

ah  ah 

Show  that  the  asj'mptotes  of  an  hyperbola  and  of  its  conju- 
gate hyperbola  are  the  same. 

Prove  that  the  distance  of  a  point  of  the  Hyperbola  from  its 
asymptote  decreases  indefinitely  as  the  point  recedes  from  the 
centre. 

32.  Prove  that  if  a  circle  be  described  on  the  transverse 
axis  of  an  ellipse  as  a  diameter,  the  ordinate  of  any  point  of 
the  ellipse  is  to  the  ordinate  of  that  point  of  the  circle  which 
has  the  same  abscissa  as  h  is  to  a. 

Prove  that  the  area  of  an  ellipse  is  irah. 

33.  Find  the  equations  of  the  tangent  and  the  normal  to  a 
conic  at  a  given  point  of  the  curve. 

[38]     Tangents:    ^  +  -M=l,  ^_M=i, 

^     -^  ""  a'       h'  a'        h' 


-y  =  a^-b-,    ■    - 
y  —  y,=  -^(x-x,). 


d          h  (r         h 

[39]     Normals:     —x yz=a^—b-,    —x-\ — y  =  a^  -i-h^, 

^1        Vi  Vi        Vi 


El 
ni 

Define  the  subtangent  and  subnormal,  and  find  their  lengths. 

^     ^                               a^  —  x,^           (^  —  x?  ^ 

[40]     Subtangents:    -,         ^,  ^^i- 


Xi 

Xy 

h-Xi 
a^ 

b^x. 

[41]     Subnormals  :     — -\  — -',  m 


34.  Prove  that  the  properties  of  poles  and  polars  referred 
to  in  sections  25  to  27  inehisive,  with  the  exception  of  the  hist 
paragi'aph  of  25,  hold  for  the  Ellipse  and  Hyperbola. 

Prove  that  the  polar  of  an}-  point  (a*,,?/i)  is  parallel  to  the 
tangent  at  the  point  where  the  line  joining  (xi^iji)  with  the 
centre  cuts  the  curve. 

35.  Find  the  distance  from  the  centre  to  each  focus  in  a 
central  conic,  and  the  equation  of  each  directrix. 

[42]     Centre  to  focus  =  ±  ae. 

Equation  of  directrix,        re  =  ±  -, 

e 


e-  = —  in  Ellipse  ; 


where 


2      a^  +  b-  . 


in  Hyperbola. 


Prove  that  the  sum  of  the  distances  of  any  point  of  the 
Ellipse,  and  the  difference  of  the  distances  of  any  point  of  the 
Hjperbola  from  the  /oct,  is  equal  to  the  length  of  the  trans- 
verse axis. 

3G.  Prove  that  the  tangent  and  normal  at  any  point  of  a 
central  conic  bisect  the  angles  between  the  focal  radii  drawn  to 
the  point. 

Prove  that  an  ellipse  and  an  hyperbola  having  the  same  foci 
intersect  at  right  angles. 

37.  Find  the  polar  of  the  focus.  Prove  that  any  chord 
through  the  focus  is  perpendicular  to  the  line  joining  its  pole 
with  the  focus. 

38.  Prove  that  the  tangent  at  any  point  of  the  Parabola 
makes  equal  angles  with  the  focal  radius  drawn  to  the  point, 
and  with  the  transverse  axis  of  the  curve. 


4G2(il8 


10 
39.   Find  the  condition  that  the  line  -  +  ^  =  1  shall  touch  the 

a       f3 
Circle  ;  the  Ellipse  ;  the  Hyperbola ;  the  Parabola. 

ma +  2/8^  =  0. 


40.   Find  the  condition  that  the  line  y=^lx  +  p  shall  touch 
the  Circle  ;  the  Ellipse  ;  the  Hj'perbola ;  the  Parabola. 

[44]     /?2  =  r2(l+Z2);     ^2^Z2a2  +  &2.     fi^^l^a'-b^; 
^     21 


41.  Find  the  equation  of  the  diameter  bisecting  a  given  set 
of  parallel  chords. 

[45]     b'^x  +  a-tan6.y  =  0\        b^x  —  a^tsine.y  =  0; 
taxid  .y  =  m. 

42 .  For  central  conies ,  prove  that  if  one  diameter  bisects  all  the 
chords  parallel  to  the  other,  the  second  will  bisect  all  the  chords 
parallel  to  the  first.  Such  a  pair  of  diameters  are  called  conju- 
gate. Show  that  the  product  of  the  tangents  of  the  inclinations 
of  a  pair  of  conjugate  diameters  to  the  transverse  axis  is  equal 

to 1  in  the  Ellipse,  and  to  —  in  the  Hyperbola.     Show  that 

of  two  conjugate  axes  only  one  can  meet  the  Hyperbola. 

Find  the  equation  of  the  diameter  conjugate  to  the  diameter 
through  (cCi,  2/1)  in  the  Ellipse  ;  in  the  Hyperbola. 

[46]     2i^+-Vll  =  0;         ^-?^  =  0. 

Conjugate  diameters  of  a  circle  are  perpendicular. 


11 

43.  Problem.  The  coordinates  of  the  extremity  of  any 
diameter  being  given,  to  find  the  coordinates  of  the  extremities 
of  the  conjugate  diameter  in  the  Ellipse  ;  in  the  Hyperbola. 

_,„_  avi  ,  bxi  ,  aifi  .  bx, 

[47]     x=^f,     y=±^;     x=±-^,     y  =  ±-^. 


44.  Prove  that  in  the  Ellipse  the  sum  of  the  squares  of  any 
two  conjugate  semi-diameters  is  equal  to  the  sum  of  the  squares 
of  the  semi-axes.  Establish  the  coiTesponding  property'  in  the 
case  of  the  Hjperbola. 

45.  Prove  that  the  product  of  the  focal  distances  of  any 
point  is  equal  to  the  square  of  the  semi-diameter  conjugate  to 
the  diameter  through  the  point. 

46.  Problem.  To  find  the  length  of  the  perpendicular  from 
the  centre  on  the  tangent  at  anj^  point  in  terms  of  the  length  of 
the  semi-diameter  conjugate  to  the  diameter  through  the  point. 

47.  Prove  that  the  parallelogram  formed  by  tangents  at  the 
extremities  of  any  pair  of  conjugate  diameters  is  equivalent  to 
the  rectangle  on  the  principal  axes. 

48.  Problem.  To  find  the  equation  of  a  central  conic 
referred  to  any  pair  of  conjugate  diameters  as  axes. 

[48]     -,±^2=1- 

49.  Problem.  To  find  the  equation  of  a  parabola  referred 
to  a  diameter,  and  the  tangent  at  its  extremity  as  axes, 

[49]     y^=2miX, 

where  m^  is  twice  the  distance  of  the  origin  from  the  focus. 


12 

50.  Problem.      To   find    the    equation    of    the    hyperbola 
referred  to  its  asymptotes  as  axes. 

[50]     4.xy  =  o?  +  h\ 

51.  Problem.    To  find  the  polar  equation  of  a  central  conic, 


of  a  parabola, 


[51]     r=<^^=^; 
1  +  e  cos  ^ 


[52]     r : 


1  +  cos  </) 


52.  Show,  by  investigating  the  general  equation  of  the  second 
degree,  that  any  equation  of  the  second  degree  can  be  reduced 
by  a  suitable  transformation  of  coordinates  to  one  of  the  forms 
already  discussed,  and  therefore  represents  some  conic. 

53.  Show  that  five  independent  conditions  are  needed  to 
determine  a  conic ;  and  solve  the  problem  of  passing  a  conic 
through  five  given  points. 

W.  E.  BYERLY, 

Professor  of  Mathematics  in  Harvard  University. 


ELEMENTS  OF  THE  DIFFERENTIAL  CALCULUS. 

Willi  Xumcrous  Examples  and  Applications.  Dosittnod  for  L'se  as  a  College  Text- 
Book.  By  W.  E.  Byeki.v,  I'rofecMor  of  Mathematics,  Harvard  University.  8vo. 
273  pages.     Mailing  Price,  $2.15  ;  Introduction,  $2.00. 

This  book  embodies  the  results  of  the  author's  experience  in 
teaching  the  Calculus  at  Cornell  and  lian'ard  Universities,  and  is 
intended  for  a  text-book,  and  not  for  an  exhaustive  treatise.  Its 
peculiarities  are  the  rigorous  use  of  the  Doctrine  of  Limits,  as  a 
foundation  of  the  subject,  and  as  preliminary  to  the  adoption  of  the 
more  direct  and  practically  convenient  infinitesimal  notation  and 
nomenclature;  the  early  introduction  of  a  few  simple  formulas  and 
methods  for  integrating;  a  rather  elaborate  treatment  of  the  use  of 
infinitesimals  in  pure  geometry;  and  the  attempt  to  excite  and  keep 
up  the  interest  ot  the  student  by  bringing  in  throughout  the  whole 
book,  and  not  merely  at  the  end,  niunerous  applications  to  prac- 
tical problems  in  geometry  and  mechanics. 

James  Mills  Pelrce,  Prof,  of  Math.,  tilic  spirit,  and  calculated  to  develop  the 

Ilarrard  UnirerxUij  (From  the  Harvard  same  spirit  in  the  learner.  .  .  .    The  book 

Register):  In  mathematics,  as  in  other  contains,  perhaps,  all  of  the  integral  cal- 

braoches  of  study,  the  need  is  now  very  cuius,  as  well  as  of  the  differential,  that  is 

much  felt  of  teaching  which  is  general  necessary  to  the  ortftnary  student.     And 

without  being  superficial ;  limited  to  lead-  with    so    much  of    this   great  scientific 

ing  topics,  and  yet  within  its  limits;  thor-  method,  every  thorough  student  of  phy- 

.  ough,  accurate,  and   practical  ;  adapted  sics,  and  every  general  scholar  who  feels 

to  the  communication  of  some  degree  of  any  interest  in  the  relations  of  abstract 

power,  as  well  as  knowledge,   but  free  thought,  and  is  capable  of  grasping    a 

from  details  which  are  important  only  to  mathematical  idea,  ought  to  be  familiar. 

the  specialist.     Prof.   Byerly's  Calculus  One  who  aspires  to  technical    learning 

appears  to  be  designed  to  meet  this  want,  must  supplement  his  mastery  of  the  ele- 

.  .  .    Such  a  plan  leaves  much  room  for  ments  by  the  Bt\idy  of  the  comprehensive 

the  exercise  of  individual  judgment;  and  theoretical  treatises.  .  .  .    But  he  who  is 

differences  of  opinion  will  undoubtedly  thoroughly  acquainted  with  the  book  be- 

exist  in  regard  to  one  and  another  point  fore  us  has  made  a  long  stride  into  a 

of  this  book.     But  all  teachers  will  agree  sound  and   practical    knowledge  of  tho 

that  in  selection,  arrangement,  and  treat-  subject  of  the  calculus.    He  has  begun  to 

ment,  it  is,  on  the  whole,  in  a  very  high  be  a  real  analyst, 
degree,  wise,  able,  marked  by  a  true  scien- 


ELEMENTS  OF  THE  INTEGRAL  CALCULUS. 

With  Numerous  Examples  and  Applications  ;  containing  a  Chapter  on  the  Calculus 
of  Imaginaries,  aud  a  Practical  Key  to  the  Solution  of  Difrerential  Equations. 
l)eslgned  for  X'.sc  as  a  College  Text-Book.  By  W.  E.  Hvkri.v,  Prof,  of  Mathematics 
in  Harvard  University.   8vo.    204  pages.    Mailing  Price,  $2.15;  Introduction,  $2.00. 

This  volume  is  a  sequel  to  the  author's  treatise  on  the  Differential 
Calculus,  and,  like  that,  is  written  as  a  text-book.  The  last  chap- 
ter, however,  —  a  Key  to  the  Solution  of  Differential  Equations, — 
may  prove  of  service  to  working  mathematicians. 

H.  A.  Newton,   Profensor  of  Mnihemalic*,    Yale  Co/leffe:  We  shall   use  it  in 
my  optional  oinss  next  term. 


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